Dynamics of Blood Flow and Oxygenation ChangesDuring Brain Activation: The Balloon Model

Richard B. Buxton, Eric C. Wong, Lawrence R. Frank

A biomechanical model is presented for the dynamic changesin deoxyhemoglobin content during brain activation. Themodel incorporates the conflicting effects of dynamicchanges in both blood oxygenation and blood volume. Calcu-lations based on the model show pronounced transients in thedeoxyhemoglobin content and the blood"oxygenation leveldependent (BOLD) signal measured with functional MAl, in-cluding initial dips and overshoots and a prolonged post-stimulus undershoot of the BOLD signal. Furthermore, thesetransient effects can occur in the presence of tight coupling ofcerebral blood flow and oxygen metabolism throughout theactivation period. An initial test of the model against experi-mental measurements of flow and BOLD changes during afinger-tapping task showed good agreement.Key words: functional magnetic resonance imaging; cerebralblood volume; cerebral blood flow; cerebral oxygen metabo-lism.

INTRODUCTION

Despite the widespread use of functional neuroimagingtechniques, the physiological changes in the brain thataccompany neural activation are still poorly understood.D~ring brain activation, a modest increase in the cerebralmetabolic rate of oxygen (CMROzJ is accompanied by amuch larger increase in local blood flow (1, 2). Because ofthis imbalance, local capillary and venous blood aremore oxygenated during activation. The large increase inflow is the basis for mapping brain activation patternswith positron emission tomography (PET), and the de-crease in local de oxyhemoglobin concentration is thebasis for functional magnetic resonance imaging (fMRI)exploiting the blood oxygenation level dependent(BOLD) effect (3-7). Recently, we proposed a possibleexplanation for the large flow increase as a result of tightcoupling of flow and oxygen metabolism in the presenceof limited oxygen delivery (8). This oxygen limitationmodel is based on the assumptions that essentially all ofthe oxygen that leaves the capillary is metabolized andthat blood flow increases are accomplished by increasedcapillary blood velocity rather than capillary recruit-

MRM 39:855-864 (1998)From the Departments of Radiology (R.B.B., E.C.W., LR.F.) and Psychiatry(E.C.W.), University of California at San Diego, San Diego, California.

Address correspondence to: Richard B. Buxton, Ph.D., Associate Professorof Radiology, UCSD Medical Center, 200 West Arbor Drive, San Diego, CA92103-8756.

Received July 3, 1997; revised November 28, 1997; accepted February 9,1998.

0740-3194/98 $3.00Copyright @ 1998 by Williams & WilkinsAll rights of reproduction in any form reserved.

MAGNETIC RESONANCE IN MEDICINE

ment. Under these circumstances, increased blood flowleads to reduced oxygen extraction due to the decreasedcapillary transit time. The rate of delivery of oxygen,which is proportional to the product of flow and oxygenextraction fraction, therefore increases much less thanthe flow itself. In the context of this model, a large in-crease in flow is required to support a small increase inoxygen metabolism.

The oxygen limitation model provides an interpreta-tion of the mismatch between flow and oxygen metabo-lism changes during brain activation as evidence for tightcoupling, rather than uncoupling, of flow and metabo-lism. However, detailed studies of the time course ofdeoxy- and oxyhemoglobin changes during activationhave revealed a complex pattern that may contain evi-dence of a transient uncoupling of flow and metabolism.In optical studies in a cat model, Malonek and Grinvald(9) found that after a brief (4-s) visual stimulation, thetotal deoxyhemoglobin increased for the first 3 s, fol-lowed after a few seconds by the larger decrease associ-ated with the BOLD effect. This "fast response" wasinterpreted as an initial increase in oxygen extraction,before the large flow increase.

In addition, there is now a substantial amount of ex-perimental data measuring the time course of the BOLDeffect. A square wave pattern of stimulus presentationtypically leads to a BOLD response that is delayed by 2 to3 s followed by a ramp of 6 to 10 s to a plateau value,followed by a return to baseline after the stimulus with asimilar ramp (10). However, the BOLD signal time courseduring brain a~1jva.tibn also has been reported to exhibitseveral transient features at the onset and end of thestimulus: an initial dip (11-13), corresponding to theinitial increase in local de oxyhemoglobin observed withoptical techniques (9); a post-stimulus undershoot notevident in):he in-flow signal (14); and an initial over-shoot followed by a much weaker plateau and a strongpost-stimulus undershoot (15, 16). A possible explana-tion for these BOLD effects is a transient uncoupling offlow and oxygen metabolism. The initial dip has beeninterpreted as evidence for an initial increase in oxygenextraction before the flow increase (9). The post-stimulusundershoot has been modeled as an elevated oxygenextraction, after flow has returned to baseline, required toreplenish depleted tissue oxygen stores (17). The initialovershoot, weak plateau, and post-stimulus undershoothave been interpreted as evidence for transient uncou-pling, recoupling, and uncoupling again of flow and ox-ygen metabolism (15, 16).

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Thus, the observed transients in signals dependent onblood oxygenation may be evidence for uncoupling offlow and metabolism. However, recent experimentalwork suggests another explanation for these phenomena.Mandeville et aI. (18, 19) found in animal studies with anintravascular marker that the dynamics of blood volumechange do not match the dynamics of the flow change.Although the deoxyhemoglobin concentration in blooddepends on flow and oxygen extraction, the total amountof deoxyhemoglobin depends strongly on blood volume,as well. For this reason, a quantitative understanding ofthe combined effects of changes in flow, oxygen extrac-tion, and blood volume is required to clarify the signifi-cance of the observed transients in the BOLD signal. Inthis article, we describe a simple biomechanical modelfor the blood volume changes during brain activation thatis consistent with tight coupling Of flow and oxygenmetabolism throughout the activation and yet producestransients similar to those observed experimentally. Apreliminary version of this work has been presented inabstract form (20).

THE BOLD SIGNAL

The mathematical model presented here describes thechanges in physiological variables during brain activa-tion. To connect the model with experimental tMRI, wefirst need a quantitative model for BOLD signal changesas a function of blood susceptibility and volume. Thisrelationship has been extensively explored in recentyears using experimental data (21), numerical MonteCarlo simulations (21-23), and analytical calculations(24). Although the signal dependence is intrinsically anonlinear function of susceptibility and blood volumedue to the effects of diffusion, the situation is relativelysimple for the case of gradient echo (GRE) signals af-fected by postcapillary blood vessels. In this initial anal-ysis, we make the simplifying assumption that the GREsignal changes are primarily due to small postcapillaryvenous vessels and neglect the contribution of capillar-ies. In this case, the role of diffusion is minor because ofthe larger size of the venous vessels, and the extravascu-lar signal changes essentially depend just on the changein the total amount of deoxyhemoglobin in the tissuevoxel (23). In addition, Boxerman et al. (25) have recentlyinvestigated the role of intravascular signal changes inBOLD experiments and conclude that at 1.5 T, thesechanges contribute more than half of the net observedsignal change. We therefore include both contributionsin our model for the BOLD signal.

As a starting point, we consider the total BOLD signalto be a volume-weighted sum of the extravascular (Se)and intravascular (Si) signal:

S = (1 - V)Se + VSi

where V is the blood volume fraction.signal changes LlS:

LlS = (1 - Vo)LlSe - LlV Se + VOLlSi + LlV Sj [2]

where V 0 is the resting blood volume fraction. Thesignalthen depends on how the intrinsic signals change when

Buxton et al.

the deoxyhemoglobin content and the volume change. Intheoretical development, we will use the notation thatcapital letters refer to a specific dimensional value of aparameter, whereas lower case letters refer to the valueduring activation normalized to the value at rest. Wewrite the normalized total deoxyhemoglobin content asq = QI~ and the normalized volume as v = V IV o' Forthe change in the extravascular component, we use thenumerical results of Ogawa et ai. (23), showing that forthe small venous vessels, the transverse relaxation rate(R2 *) is proportional to the product of the blood deoxy-hemoglobin concentration and blood volume, so ~Se de-pends only on the total deoxyhemoglobin q. For theintravascular component (and for the intrinsic ratioS/Se)' we use the numerical results of Boxerman et ai.(25). From their calculations (their Fig. 2), ~Si can bereasonably approximated as a linear function of the de-oxyhemoglobin concentration in blood for O2 extractionfractions between 20% and 50% (the range of interest forbrain). Combining these numerical results with Eq. [2]leads to the following form for the BOLD signal (seeAppendix for details):

~ss = Vo[ k1(1 - q) + k2( 1 - ;) + k3(1 - V)] [3]

The first term describes the intrinsic extravascular signal,the second term describes the intravascular signal, andthe third term describes the effect of changing the bal-ance of the sum in Eq. [1]. The parameters kl' kz' and k3are dimensionless and can be estimated from the earlierwork. For Bo = 1.5 T and TE = 40 ms, we estimate thatk[ ~ 7 Eo based on the numerical studies of Ogawa et 01.(23), where Eo is the resting oxygen extraction fraction.Based on the results of Boxerman et 01. (25), we estimatekz ~ 2 and k3 ~ 2 Eo - 0.2. In Eq. [3], q is the normalizedtotal voxel content of deoxyhemoglobin (Le., q = 1 atrest), v is the normalized venous blood volume (v = 1 atrest), and V 0 is the actual venous blood volume fraction(e.g., 1-4%).

BALLOON~MODEL

To model tile transient aspects of the BOLD signal, weassume again that there is no capillary recruitment, inkeeping with the oxygen limitation model above, so thatblood volume changes occur primarily in the venouscompartment. (The arteriolar dilitation that produces aflow increase is assumed to be a negligible change inblood volume.) The vascular bed within a small volumeof tissue is then modeled as an expandable venous com-partment (a balloon) that is fed by the output of thecapillary bed. The volume flow rate (ml/s) into the tissue,Fin(t), is an assumed function of time that drives thesystem. The volume flow out of the system, F out(t), isassumed to depend primarily on the pressure in thevenous compartment. For example, we could write thatF out = (P - P mixed)/R, where P is the pressure in thevenous compartment, P mixed is the mixed venous pres-sure downstream from the tissue element, and R is theresistance of the vessels after the balloon. Then, the phys-ical picture of activation is that after the arteriolar resis-

Balloon Model for BOW Dynamics During Brain Activation

tance is decreased, producing an increase in Fin' thevenous balloon swells, and the pressure increases untilF out matches Fin (a new steady state). The amount ofswelling that occurs will depend on the biomechanicalproperties of the vessel as reflected in the pressure/vol-ume curve of the venous balloon. However, rather thanintroduce pressures explicitly into the equations (andthus introduce more parameters), we simply assume thatF out is a function of the venous volume, V. The rate ofchange of the volume of the balloon is the differencebetween Fin and F out' We can then consider the totaldeoxyhemoglobin Q(t) in the tissue element. For simplic-ity, we neglect the capillary contribution and assume thatall of the de oxyhemoglobin is in the venous com part-ment. The rate of entry of de oxyhemoglobin into thevenous compartment is Fin E Ca, where E is the netextraction of O2 from the blood as it passes through thecapillary bed and Ca is the arterial O2 concentration(assumed to be due to a fully oxygenated hemoglobinconcentration), If we then treat the balloon as a well-mixed compartment, the clearance rate of deoxyhemo-globin from the tissue is F out X the average venous con-centration, Q(t)/V(t). The coupled equations for Q(t) andV(t) are then

Q(t)dQ = Fin(t)ECa - FouI(V) V(t)

dt

[4]

dV- = Fin(t) - Fout(V)dt

By scaling each of these variables with their value at rest(t = 0), these equations can be written as in the previoussection in terms of the dimensionless variables q(t) -

Q(t)/~, v(t) = V(t)/V 0' fin(t) = Fin(t)/F 0' and faut(v) =F aut(V)/F 0:

dq 1 [ E(t) q(t) ]- = - fin(t)- - fout(v)-

dt To Eo v(t)

where ~ is the resting deoxyhemoglobin content, V 0 isthe resting volume, Fo is the resting flow, 'To = Vo/Fo isthe mean transit time through the venous compartment atrest, Eo is the resting net extraction of O2 by the capillarybed, and q(O) = v(O) = fin(O) = fout(v(O)) = 1. Note that 'Tosimply sets the time scale for changes and that the onlyother parameter that appears explicitly is Eo. However,there are two functions that remain to be specified: E(t)and fout(v). At this point, we can include the oxygenlimitation model by requiring that the net extraction isequal to the unidirectional extraction. That is, we assumethat CMR02 increases as much as possible within theconstraints set by limited oxygen delivery. Under thesecircumstances, we argued previously that a nonlinearexpression for E(f) is a reasonable approximation for awide range of transport conditions (8):

E(f) = 1 - (1 - Eop/f [6]

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In Eq. [5], E(t) is then replaced with E(fin(t)) from Eq. [6].Variable temporal patterns of response will then be cre-ated for different functional forms of fout(v) , which isequivalent to different pressure/volume curves for thevenous balloon.

In constructing fout(v), we need to distinguish betweenthe steady-state relationship of blood flow and volumeand the transient relation fout(v) that controls the transi-tion from one steady state to another. In early work usingaltered CO2 to change blood flow, Grubb et 01. (26) foundthat the steady-state total blood volume could be de-scribed empirically by a power law relationship, v = fa,where lX was found to be about 0.4 from their data. Thesimplest assumption would be that fout(v) follows thissame empirical relation. However, experimental mea-surements by Mandeville et 01. (18, 19), using a vascularcontrast agent, suggested that although the steady-staterelationship was in reasonable agreement with Grubb et01. (26), the transition periods were not. Therefore, for thenumerical studies described here, we assumed that thesteady-state plateau level of volume change is given by apower law with exponent lX, but the time course duringthe transition between steady-state levels is different. Inpractical terms for the calculations, this means that theend points of the curve fout(v) are specified by the powerlaw relation, but the shape of the curve between thoseend points is variable. For the calculations, we modeledfout(v) as a sum of a linear component and a power law.

time course of cerebral blood flow and BOLD signalchanges during a simple finger-tapping task. Measure-ments were made with a new pulsed arterial spin label-ing technique recently developed in our laboratory,called Quantitative Imaging of Perfusion with a SingleSubtraction, version II (QUIPPS II) (27, 28). Arterial spinlabeling involves tagging the arterial blood with an incversion pulse before entering the slice of interest, a delayto allow blood to enter the slice, followed by collection ofa rapid echo planaF image, creating the tagged image. Thepulse sequence ts th~n repeated without the initial inver-sion to create a control image. The difference image,control minus tagged, is then directly proportional to theamount of blood delivered during the delay period andthus is directly proportional to local perfusion. QUIPSSII is specifically designed to deal with the confoundingeffects of nonuniform transit delays from the taggingregion to different brain regions, which can severelycompromise perfusion estimates based on measurementswith a single inversion delay. In QUIPSS II, at time TIiafter the inversion pulse, a saturation pulse is applied tothe tagging region, destroying that portion of the taggedblood that remains in the tagging region and thus forcingthe time width of the tag to equal TIi. The image is thenacquired at a later time, TIz, to allow the tagged bolus ofblood to reach all of the voxels in the imaged slice.Provided that the delay TIz-Tli is longer than the largesttransit delay, then all of the tagged blood destined foreach voxel will have arrived.

[5]

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For the current studies, we used an off-resonance con-trol RF pulse, identical to the inversion pulse, but withno slice selective field gradients (PIC ORE) (28), to pro-duce similar off-resonance effects on the imaged slicewithout tagging arterial blood. We used a GE 1.5-TSIGNA clinical MRI scanner (General Electric Corp.,Milwaukee, wi) fitted with high performance localhead gradient and RF coils of our own design and con-struction (29). Pulse sequence parameters for these gra-dient echo acquisitions were TI1 = 600 ms, TIz = 1200 ms,TE = 30 ms, TR = 2.011 s, 64 X 64 echo-planar imag-ing acquisition with in-plane resolution 3.75 X 3.75 mmand slice thickness 8 mm. Repetition times on the orderof 2 s are required to allow replenishment of fully relaxedblood. To increase our temporal resolution, the TRwas chosen to be slightly asynchronous with the stimu-lus period so that when multiple cycles are collapsed toform an average single cycle time course, later cyclesprovide measurements in-between the measurements.ofthe first cycle, producing a time resolution of approxi-mately 1 s.

Six healthy volunteers were asked to perform bilateralsequential finger tapping for 1 min, followed by 2 min ofrest. This 3-min cycle was repeated eight times, while750 images of a single axial slice passing through themotor areas were collected alternating between tag and- . - -control images. In the raw datatime course, the flow signal isencoded in the image-to-imagedifference signal, and theBOLD signal is encoded in theslow variations of the averagesignal after the stimulus. Theimages are alternately posi-tively and negatively flow-weighted by the arterial tag. Toremove the flow weightingfrom the BOLD signal, eachimage is added to the averageof its two neighbors in time. Tominimize the BOLD weightingin the flow signal, each imageis subtracted from the averageof its neighbors. Each timecourse is then collapsed appro-priately over cycles, to pro-duce an average time coursefor one cycle for each pixel.For each subject, two 8 X 8block of pixels covering theleft and right motor areas wereselected for further analysis.From these 128 pixels per sub-ject, all pixels showing achange with activation in theflow signal of >40% were se-lected, and the flow and BOLDtime courses for these pixelswere averaged across all sub-jects (a total of 153 pixels) toproduce high signal-to-noisedynamic curves. In the re-

Buxton et al.

ported data, the amplitude of the flow signal is plotted asa percentage of the total mean brain signal. Absolutequantitation in units of cerebral blood flow (CBF) re-quires a global scaling factor that depends on the protondensity and relaxation times of blood. This calibrationfactor was not measured, but we estimate for assumedrelaxation parameters that a 1 % flow signal correspondsto CBF of 50 to 60 ml/min-ml of tissue.

RESULTS

The Cl}rves in the figures illustrate the variety of timecourses that can be produced by the balloon model.These curves were calculated by numerical integration ofEq. [5] using the Mathematica software package (WolframResearch). For all of these calculations, To = 2 sandEo = 0.4, and the driving function ofthe system fin(t) was

assumed to be a trapezoidal function with a rise time of4 to 6 s and variable duration. These calculations de-scribe the response of the system to a given inflow curve.That inflow curve, in turn, is presumed to be a responseto a particular neural stimulus. However, the relation-ship between stimulus and flow response is poorly un-derstood, and in our calculations, we do not attempt tomodel it. In this section, we will refer to the "duration" of

FIG. 1. Balloon model curves for a linear form of fouiv), showing a simple pattern of response. Thescale for the BOLD plot is percent signal change, and this amplitude scales directly with theblood volume Vo. Additional model parameters in these calculations were Eo = 0.4, Vo = 0.01, anda = 0.5.

Balloon Model for BOLD Dynamics During Brain Activation

expected finding is discussedin more detail below. Despitethese pronounced transientfeatures, CMROz is tightlycoupled to fin(t) throughoutthe activation period.

Figure 3 shows a compari-son of the experimental datawith balloon model calcula-tions. The measured flow sig-nal indicates an approxi-mately 70% increase in flowand shows a roughly trapezoi-dal shape. The BOLD datashow a similar shape, but witha pronounced post-stimulusundershoot that is not presentin the flow signal. The ob-served discrepancy betweenthe flow and BOLD signalsconfirms earlier studies of thepost-stimulus undershoot (14).The balloon model producescurves that are very similar tothe experimental data for themodel parameters and fout(v)curve shown in Fig. 2. Themodel BOLD curve also showsa small initial dip, but the tem-poral resolution and signal-to-noise ratio of the experimentaldata are not sufficient to detect

FIG. 2. Balloon model curves for a nonlinear form of fou.(v), which produces a more complex pattern such a small signal change.of transients. Outflow lags behind inflow, producing an initial fast rise in blood volume and a slow Figure 4 shows more spe-return to baseline. The total deoxyhemoglobin content briefly increases (the fast response), then cifically how the post-drops sharply, and finally overshoots the resting level after the flow has returned to the resting stimulus undershoot comeslevel. The BOLD response is qualitatively the inverse of the total deoxyhemolgobin response, b t' th t t f th b 1showing a weak initial dip, followed by a signal increase, and ending with a prolonged post- a ou III e con ex 0 e a-stimulus undershoot. Note, however, that the initial dip is weaker than the post-stimulus under- loon model. The undershootshoot despite the fact that the total deoxyhemoglobin change is slightly larger during the fast occurs when the volume re-response. Inflow and CMROz do not show any of the transients seen in the deoxyhemoglobin turns to baseline more slowlyor BOLD signals. Additional model parameters in these calculations were Eo = 0.4, Vo = 0.01, and than flow, producing an ele-ex = 0.5. vated total deoxyhemoglobin

just due to an increased vol-an activation as the time from the beginning of the up ume of blood, even though the blood oxygenation hasramp of fin(t) to the beginning of the down ramp. In each returned to baseline. The rate at which the excess bloodfigure, the time courses for different dynamical quanti- volume clears from the tissue depends on the difference

.J:ies are shown in separate labeled panels. between outflow and inflow. Thus, a prolonged post-Figure 1 shows the calculated response to a short, 8-s stimulus undershoot occurs when font returns nearly to

duration stimulus with a maximum flow change of 70% baseline while the volume is still high. More specifically,when fout(v) is a linear function. Outflow lags slightly the duration of the undershoot depends inversely on thebehind inflow, but all of the curves show a simple pat- initial slope of fout(v). Figure 4 shows that a very subtletern. In Fig. 2, the only change was to make fout(v) a change in the initial slope has a strong effect on thestrongly nonlinear function, which produces much more undershoot duration.complex transient behavior. There is an initial sharp Figure 5 illustrates the dynamics ofthe fast response inincrease in total de oxyhemoglobin (the fast response), the context of the balloon model. To emphasize thesefollowed by a sharp decrease, and then an overshoot effects, a short inflow duration of 4 s was used (I.e., a 4-sabove baseline. The BOLD signal shows the correspond- up ramp immediately followed by a 4-s down ramp). Theing inverse pattern, including an initial dip and a pro- model parameters and form of fout(v) were the same as inlonged post-stimulus undershoot. Note, however, that Figs. 2 and 3. The curves show an initial increase inthe initial dip is much weaker than the post-stimulus deoxyhemoglobin, peaking around 2.5 s after the begin-undershoot, despite the fact that the corresponding ning of the flow up-ramp, followed by a decrease inchanges in total deoxyhemoglobin are similar. This un- de oxyhemoglobin that reaches a minimum at about 6 s.

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FIG. 3. Experimental data measured with an arterial spin labeling method (OUIPSS II) that allowssimultaneous measurement of flow and BOLD signals (top), compared with the balloon modelcurves (bottom) for the same model parameters used in Fig. 2.

The precise quantitative features of this curve, such asthe relative amplitudes of the positive and negative de-oxyhemoglobin changes, depend on the specific modelparameters and the assumed inflow function. Neverthe-less, this temporal pattern is similar to that found byMalonek and Grinvald (9) in optical studies in a catmodel with visual stimulation. They found an initialde oxyhemoglobin peak approximately 2 s after the begin-ning of the visual stimulus, which then decreased to aminimum at about 6 s. In Fig. 5, the initial rise in totaldeoxyhemoglobin is primarily due to increased bloodvolume rather than altered blood oxygenation. As flowcontinues to increase, the blood oxygenation increasesand eventually dominates over the volume change, re-ducing the total deoxyhemoglobin.

Figure 5 also provides an explanation for why theinitial dip in the BOLD signal is weaker than the post-stimulus undershoot, despite the fact that the deoxyhe-moglobin changes are similar. The lower three panelsshow separate plots of the intrinsic signal contributionfrom the intra- and extravascular BOLD signals and thenet BOLD signal, which is a weighted average of the othertwo. The fast response is dominated by a volume change,

Buxton et al.

which increases total deoxy-hemoglobin and creates a dipin the extravascular BOLD sig-nal. However, the intravascu-lar signal shows a smooth in-crease due to the steadilyincreasing blood oxygenationand is contributing a moder-

ately strong positive signalchange at the time of the dipwhen the extravascular signalchange is negative. Becausethese two signals contributeroughly equally to the net sig-nal at 1.5 T, the dip is nearlycancelled in the net BOLD sig-nal. At higher fields, the BOLDsignal is likely dominated bythe extravascular BOLD effect,so the dip may be more pro-nounced. This may partly ex-plain, in addition to signal-to-noise considerations, why theinitial dip is rarely seen atlower fields. In contrast, dur-ing the post-stimulus under-shoot, the intrinsic intravascu-lar signal has returned tobaseline because blood oxy-genation has returned to base-line, and therefore does notconflict with the undershootof the extravascular signal.

Figure 6 illustrates the ef-fects of rapidly cycling thestimulus presentations. In thisexample, a 10-s duration in-flow pattern is cycled with aperiod of 20 s. Thus, the re-

peated stimuli occur during the post-stimulus under-shoot of the previous stimulus. As a consequence, theblood volume never returns to baseline, and the apparentbaseline of the BOLD signal is lowered. In addition, thecurve of total deoxyhemoglobin is altered in a way thatalso looks like a shift of the baseline.

In Figure 7, we extended the simple forms of fout(v)used in the other figures to try to produce time coursessimilar to those found by Frahm and coworkers (15, 16).They reported an initial overshoot in the BOLD signal,followed by a slow decrease to a modest (or weak) pla-teau level, followed by a pronounced post-stimulus un-dershoot. The calculations in Fig. 7 differ in two waysfrom the previous figures: (1) the blood volume change islarger (a = 0.7), and (2) there is hysteresis in the curve of

fout(v). Given the viscoelastic nature of blood vessels, it islikely that the pressure/volume curve during inflationmay be different from that during deflation due to stressrelaxation effects. In Fig. 7, fout(v) during inflation in-creases rapidly with increasing volume, but shows acurve similar to that used in the previous figures at theend of the stimulus. The behavior of the curves at thebeginning ofthe inflow increase is opposite to that in Fig.~

Balloon Model for BOW Dynamics During Brain Activation

FIG. 4. Curves for a 1 O-s duration of inflow, illustrating the post-stimulus undershoot. The durationof the undershoot depends inversely on the initial slope of the curve fout(v) (left panel). The slightly

steeper slope (solid line) causes the volume change (middle panel) and the BOLD signal under-shoot (right panel) to resolve more quickly, while the shallower slope (dashed line) leads to a muchlonger undershoot.

5. Because oJ the "stiffness" of the balloon during theonset of the flow change, the volume increases moreslowly, and the change in deoxyhemoglobin is domi-nated by the change in blood oxygenation. The bloodoxygenation reaches its new steady-state value before thevolume has reached its new steady-state value, and dur-ing this gap, the total deoxyhemoglobin reaches its min-imum value, corresponding to the initial overshoot in theBOLD signal. Note that in the context of the balloonmodel, such pronounced transient effects can occur inthe presence of tight coupling of flow and oxygen metab-olism.

DISCUSSION

In recent years, studies of the local changes in deoxyhe-moglobin content associated with brain activation havefound several transient effects that may be evidence foruncoupling of cerebral blood flow and oxygen metabo-lism. These features include initial dips and overshootsand a prolonged post-stimulus undershoot in the BOLDbut not the flow signal. The earliest attempt to accountfor these transient features of the BOLD signal timecourse with a biophysical model that we are aware of isthe work of Davis et al. (17). They modeled the capillaryas a distributed compartment and the venous vasculatureas a single well-mixed compartment. Model calculationsshowed an initial dip and for some combinations ofparameters a post-stimulus undershoot, as well, althoughthe time constant for the undershoot was shorter thanwhat is usually observed. In the context of their model,O2 is normally delivered in excess of what is needed formetabolism, so there is a resting non-zero O2 concentra-tion in the tissue. The initiaJ dip is then due to an

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immediate increase of O2 me-tabolism before flow increasesby metabolizing this tissuepool of oxygen. Malonek andGrinvald (9) proposed asimi-lar interpretation for the tran-sient rise in deoxyhemoglobin.In the Davis model, the post-stimulus undershoot occurswhen the increase in CMR02is larger than the increase indelivery of O2 by flow (theproduct of the unidirectionalextraction and flow). Then, alarger fraction of the initiallyextracted O2 is metabolized, sothat the tissue concentration ofOz drops. The post-stimulusundershoot then reflects con-tinued extraction of Oz to re-plenish the tissue oxygen poolafter CMROz has returned tonormal. However, the explana-tion of the observed effects interms of this model has twoproblems. First, the modeldoes not appear to account forthe long time constants ob-

served for the undershoot. Second, blood volumechanges are added on at the end, rather than being incor-porated as an effect of flow changes. The Davis modelalso appears to be incompatible with the oxygen limita-tion model, which assumes that flow and Oz metabolismare tightly coupled because all of the extracted Oz ismetabolized and there is no reserve pool of Oz in tissue todraw on. This view is supported by the low tissue POzvalues measured experimentally (30) and also by recentdirect measurements of limited oxygen extraction (31).

To understand these various features of the BOLD timecourse, we have developed a biophysical model that ismotivated by the Davis model but incorporates the oxy-gen limitation model (8) and blood volume changes asintegral parts. The'E1ssential feature of the balloon modelis that we assume that the capillary volume is fixed butthe venous volume can change (like a balloon) with apressure/volume response curve that may vary. In thisinitial development of the model, the shape of this curveis left as an unknown function, and Eq. [5] then followsas a consequence of mass balance. To connect the dy-namic variables with measured BOLD signal curves, weused an expression for the MR signal based on recenttheoretical calculations that should be appropriate forpostcapillary vessels. Specifically including capillary ef-fects, taking account of varying oxygen saturation alongthe capillary would improve the estimates of the con-stants k1, kz, and k3' As these relationships become betterunderstood through future experimental and theoreticalwork, these values should be revised, but the qualitativefeatures of the model curves are not strongly sensitive tothe precise values of the ks. The model curves shown inthis article are meant to indicate the range of dynamicbehavior that the balloon model can produce, and the

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..~. ..~.. .~~ ..~..~.~., ...~

Flow Deoxy-Hb model is a concrete mathemat-1.5 ical representation of how

blood volume changes in con-1.04 junction with blood oxygen-

1.4 \ 1.3 ation changes can create com-\ 1.02 plex dynamics in the BOLD

1.3 \ signal. One unexpected result\ 1 1.2 was that the same model pa-

1.2 \ rameters that provide a good\ 0.98 description of our BOLD data\ 1.1 measured during finger tap-\ 096 ping in humans (Figure 3), for\ . a short duration stimulus, pro-

1 _-m 1 duce a pattern of deoxyhemo-0 10 0 . 10 0 10 globin change that is similar to

ex-vasco BOLD in-vasco BOLD net BOLD (%) previously published optical0 4 measurements in cat during0 1 . 0.4 visual stimulation (9) (Fig. 4).

. This similarity should be in-

0.05 0.3 0.3 terpreted with caution, be-cause the optical studies had a

0 0.2 spatial resolution of 50 p,m0.2 and large veins were excluded,

-0.05 0 1 whereas the fMRI data had a0.1 . spatial resolution of several

-0.1 millimeters. Furthermore, the0 experimental data presented

0 here only show the post-stim-0 10 0 10 0 10 ulus undershoot clearly; the

temporal resolution and sig-Time (sec) nal-to-noise ratio were not suf-

.t:_,--+ +- __1'_\..1.. ..:1_+--+ +\..-ficient to reliably detect the

FIG. 5. The structure of the fast response in the deoxyhemoglobin, oxyhemoglobin, and BOLD fast response. Nevertheless,signals. For this simulation, the in-flow increases linearly for 4 s then decreases linearly for 4 s. these comparisons with exper-Outflow lags behind inflow by several seconds. In the bottom row, the intravascular, extravascular, imental data suggest that theand net BOLD signals are plotted separately. In this model, the initial increase in deoxyhemoglobin model captures some of the es-is due primarily to a blood volume increase, rather than a change in the concentration of sential features of the hemody-deoxyhemoglobin in the blood. For this reason, only the extravascular component shows the initial namic response.dip. The conflict between the intravascular and extravascular signals reduces the dip in the net The physical origin of theseBOLD signal, which is a weighted sum of the two. These calculations are based on modeling of the tr . t f£ t . th d Iintravascular signal for 1.5 T, where it contributes about two-thirds of the total signal change. At . a:l~n th e ~c s ill t ~o £ ehigher fields, the intravascular contribution is reduced relative to the extravascular signal, and so)he ~ a ~ ~~ c~ns an ~ orinitial dip should be more pronounced. Model parameters for these curves are the same as in.Fig. 2. ow an 00 vo ume

changes are not the same, andrecent experiments in animal

precise shapes of the curves will vary with a different models support this scenario (18, 19) for the post-stimu-choice of parameters (e.g., a different shape of fout(v)). Ius undershoot. As illustrated here, a simple biomechani-Three basic patterns of BOLD signal transients are de- cal model can produce blood volume changes that re-scribed by the model: (1) an initial dip and a correspond- solve very slowly, depending on the shape of the outflowing increase in de oxyhemoglobin occur when inflow ex- curve as a function of volume. At this stage of develop-ceeds outflow and the blood volume rapidly increases; ment of the model, this curve is an arbitrary, but physi-(2) an initial overshoot occurs when blood volume in- cally plausible, mathematical function. A natural exten-creases more slowly than inflow; and (3) a post-stimulus sion of the model is a more complete biomechanicalundershoot occurs when blood volume returns to base- model of the viscoelastic properties of the vessels, so thatline more slowly than inflow. In addition, the model the outflow curve could be expressed in terms of param-predicts that the initial dip should be weak at 1.5 Teters related to these physical properties. Such a modelbecause of conflicting signal changes in the extravascular would then allow estimates of these physical propertiesand intravascular signals. from measured flow and BOLD time courses.

The balloon model is undoubtedly an overly simplistic These numerical calculations suggest that measure-biophysical model of the hemodynamic response to ac- ments of deoxyhemoglobin content are not reliable indi-tivation in the brain. This is in part due to the fact that cators of flow and oxygen metabolism changes withoutthe biomechanical properties of the brain vasculature are taking into account the volume changes as well. Pro-

Buxton et al.

not well known. However, themodel is a concrete mathemat-ical representation of howblood volume changes in con-junction with blood oxygen-ation changes can create com-plex dynamics in the BOLDsignal. One unexpected resultwas that the same model pa-rameters that provide a gooddescription of our BOLD datameasured during finger tap-ping in humans (Figure 3), fora short duration stimulus, pro-duce a pattern of deoxyhemo-globin change that is similar to

previously published opticalmeasurements in cat duringvisual stimulation (9) (Fig. 4).This similarity should be in-terpreted . with caution, be-

cause the optical studies had aspatial resolution of 50 /-Lmand large veins were excluded,whereas the tMRI data had aspatial resolution of severalmillimeters. Furthermore, the

Time (see)

Balloon Model for BOLD Dynamics During Brain Activation

nounced transient features of the BOLD signal can resultfrom differences in the separate time courses of flow andvolume changes. Furthermore, these transient changescan occur even in the presence of tight coupling of flowand oxygen metabolism throughout the activation pe-riod.

APPENDIX

Derivation of the MR Signal Equation

We model the MR signal as a volume-weighted sum ofthe intrinsic extravascular signal Se and the intravascularsignal Si' represented in Eq. [1]. A linear variation, validfor small changes, leads to an expression for the signalchange, Eq. [2]. By factoring out V 0 and Se from these

863

FIG. 6. Effects of multiple stimilus cy-cles. If the stimulus is repeated be-fore the post-stimulus undershoot isresolved, the blood volume does notreturn to the resting value, and theapparent baseline in the BOLD signaltime course is lowered. Note alsothat the shape of the deoxyhemoglo-bin response extracted from one cy-cle in the middle of a string of cyclesis different from the initial cycle. Forexample, the initial cycle (t = 0-20 s)

shows a large increase, followed by adecrease about three times larger,followed by an overshoot of thebaseline. However, the second cycle(t = 20-40 s) appears to be an ele-

vated baseline followed by a weakerincrease, a strong decrease, and areturn to the elevated baseline.

equations, the fractional signal change can then bewritten as:

LlS Vo

s 1 - V 0 + {3V 0

[A1]

where 13 = S/Se is the intrinsic signal ratio at rest, and vis the blood volume normalized to the volume at rest.Ogawa et a1. (23) found by numerical simulations that,for larger vessels, the gradient echo transverse relaxationrate is approximated by Rz * = 4.3 N, where f = ~XUJoE is

the susceptibility difference between the intra- and ex-

FIG. 7. The effect of hysteresis inthe form of fouiv). In this calcula-tion, the outflow as a function of vshown in the upper right was differ-ent bn expansion and contraction(as indicated by the arrows),roughly approximating a balloonthat is initially stiff, but after pro-longed expansion is more compli-ant. In addition, the volume changein this example is larger to illustratethe type of effects that can arise.The BOLD signal shows a stronginitial overshoot, followed by agradual return to a reduced pla-teau, and ending with a strongpost-stimulw, undershoot, similarin pattern to experimental data re-

ported by Frahm and coworkers(15, 16). Inflow and CMRO2 arecoupled throughout and show asimple trapezoidal pattern. Addi-tional model parameters in thesecalculations were Eo = 0.4, Va = 0.02,and a = 0.7.

864

travascular spaces expressed as a frequency, V = vV 0 isthe blood volume fraction, and E is the O2 extractionfraction. To put this into our notation with total deoxy-hemoglobin q and volume v normalized to their values atrest, it is helpful to note that E/Eo = q/v. Then thetransverse relaxation rate is R2 * = 4. 3 ~XwoEo V Qq. The

change in the extravascular signal is then:

AS.- = - AR2*TE = aVo(1- q) [A2]S.

with a = 4.3 ~XCl)oEoTE. For 1.5 T and ~X = 1 ppm, ~XCI)o= 40.3 S-1. and with resting extraction Eo = 0.4 and TE =40 ms, the scaling factor is approximately a = 2.8.

For the intravascular signal change and the restingsignal ratio {3. we used the results of Boxerman et a1. (25).In their numerical simulations for 1.5 T. TE'; 40 ms, andvessel radius = 25 ""ill, they calculated the attenuationfactor A of the intravascular component as a function ofthe fractional oxygen saturation of hemoglobin. Based ontheir results (their Fig. 2), the attenuation factor can beapproximated as A = 0.4 + 2(0.4 - E) in the range 0.15 <

E < 0.55. Assuming that the full intra- and extravascularsignals are the same in the absence of deoxyhemoglobin,the resting signal ratio is {3 = A, so for Eo = 0.4, {3 = 0.4.The intravascular signal change is then:

Combining Eqs. [AI] to [A3] yields Eq. [3], and becausethe factor multiplying Eq. [AI] is approximately V 0 forsmall resting blood volume fractions, the coefficients arek1 = a = 2.8, k2 = 2, and k3 = 0.6.

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